Symmetries of (quasi)periodic materials: Superposability vs. Indistinguishability

• An adapted notion of material symmetries for quasiperiodic media is defined • Indistinguishability generalizes the classical notion of superposability • Symmetry properties of periodic and quasiperiodic media are expressed in Fourier space • Point groups and symmorphism are computed by image processing in Fourier space • Penrose tilings exhibit tenfold rotational symmetry This work is devoted to the study of the symmetries of (quasi)periodic architectured materials. The main goal is to extend the symmetry criterion of superposability for strictly periodic to quasiperiodic materials. For this purpose, the weaker symmetry criterion of indistinguishability is used. It relies on a statistical description of the mesostructure and is defined in terms of the spatial autocorrelation functions of the material under consideration. By using the representation of these autocorrelation functions in Fourier space, the space groups of both periodic and quasiperiodic materials can be obtained. In this context, an image-processing methodology is proposed to identify the key characteristics of a material’s space group (i.e its point group and its symmorphism) directly from the Fourier transform of the mesostructure. The method is validated on synthetic two-dimensional images of (quasi)periodic architectured materials and it is pointed out, as an illustrative example, that the rotational symmetry of the classical Penrose tiling is of order ten.